Problem 86 - Elementary and Intermediate Algebra

Question

Solve each equation. Write all proposed solutions. Cross out those that are extraneous. $$ \left(m^{2}-12 m-3\right)^{1 / 2}=\left(m^{2}+12 m+3\right)^{1 / 2} $$

Step-by-Step Solution

Answer

The solution is \(m = -\frac{1}{4}\), and it is not extraneous.

Step 1: Understand the Problem

We are given two square root expressions equal to each other: \(\sqrt{m^2 - 12m - 3} = \sqrt{m^2 + 12m + 3}\). We need to solve for \(m\).

Step 2: Remove the Square Roots

To eliminate the square roots, square both sides of the equation. This results in \((m^2 - 12m - 3) = (m^2 + 12m + 3)\).

Step 3: Simplify the Equation

Simplify the equation by subtracting \(m^2\) from both sides, which cancels \(m^2\) out. The equation becomes \(-12m - 3 = 12m + 3\).

Step 4: Solve for m

Add \(12m\) to both sides to combine like terms: \(-12m + 12m - 3 = 12m + 12m + 3\). Simplifying gives \(0 - 3 = 24m + 3\). Then subtract 3 from both sides: \(-6 = 24m\). Finally, divide by 24: \(m = -\frac{1}{4}\).

Step 5: Verify the Solution

Substitute \(m = -\frac{1}{4}\) back into the original equation to check for extraneous solutions. First calculate each expression: \(\sqrt{\left(-\frac{1}{4}\right)^2 - 12(-\frac{1}{4}) - 3}\) and \(\sqrt{\left(-\frac{1}{4}\right)^2 + 12(-\frac{1}{4}) + 3}\). Simplify both roots.

Step 6: Simplify and Compare

Calculate the terms: \(-\frac{1}{4}^2 = \frac{1}{16}\), \(-12 \times -\frac{1}{4} = 3\), \(12 \times -\frac{1}{4} = -3\). Therefore, the first square root becomes \(\sqrt{\frac{1}{16} + 3 - 3} = \sqrt{\frac{1}{16}}\), which simplifies to \(\frac{1}{4}\). The second square root becomes \(\sqrt{\frac{1}{16} - 3 + 3} = \sqrt{\frac{1}{16}}\), which is also \(\frac{1}{4}\). Both expressions are equal.

Key Concepts

Extraneous Solutions

When solving equations, especially those involving square roots, it is crucial to watch out for what are known as extraneous solutions. These are solutions that, while they may appear valid as you solve through algebraic manipulations, don't actually satisfy the original equation. This situation can often arise when both sides of an equation are squared to eliminate square roots.

Squaring both sides of an equation is a common technique, but it may introduce solutions that do not actually work because they only satisfy the squared version of the equation, not the original one. Therefore, after finding potential solutions, it’s important to substitute each back into the original equation to check their validity.

To spot extraneous solutions:

• Substitute back each proposed solution into the original equation.
• Calculate each side independently to see if they are equal.
• If they aren't equal, the solution is extraneous and must be disregarded.

In our case, after solving the square root equation, we verified that the solution did indeed satisfy the original equation, so no extraneous solutions were present.

Equation Simplification

Simplification is a critical step in solving equations, especially when dealing with square roots. Simplification generally means combining like terms and reducing the equation to its simplest form. This often makes the remainder of the problem easier to solve.

In the given square root equation, \[ \sqrt{m^2 - 12m - 3} = \sqrt{m^2 + 12m + 3} \]our first move was to eliminate the square roots by squaring both sides. This results in the expressions beneath the roots becoming equal:

• (m^2 - 12m - 3) = (m^2 + 12m + 3).

We then subtracted the square term, \(m^2\), from both sides. This operation effectively cancelled the terms and revealed a much simpler linear equation:\[-12m - 3 = 12m + 3.\]

Through further simplification steps, such as adding terms and dividing by coefficients, we isolated the variable \(m\). Simplification helps in breaking down complex equations into manageable pieces, ultimately leading to easier and clearer solutions.

Algebraic Verification

After finding a solution, it is always necessary to confirm its correctness through algebraic verification. This step ensures that no miscalculations were made along the way and that the solution truly satisfies the original equation.

For our equation, we proposed the solution \(m = -\frac{1}{4}\). Verification involves substituting this value back into the original squares of the equation:

• Calculate the inner expressions: \(\left(-\frac{1}{4}\right)^2 - 12\left(-\frac{1}{4}\right) - 3\) and \(\left(-\frac{1}{4}\right)^2 + 12\left(-\frac{1}{4}\right) + 3\).
• Compute each side of the equation separately to check for equality.

Here, both roots simplify to \(\frac{1}{4}\), confirming our solution. Algebraic verification is crucial as it validates the solution and rules out any computational errors, ensuring the results are precise and accurate.